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In [[mathematics]], an '''eigenplane''' is a two-[[dimension]]al [[invariant subspace]] in a given [[vector space]]. By analogy with the term ''[[eigenvector]]'' for a vector which, when operated on by a [[linear transformation|linear operator]] is another vector which is a [[scalar (mathematics)|scalar]] multiple of itself, the term '''''eigenplane''''' can be used to describe a two-dimensional [[plane (mathematics)|plane]] (a ''2-plane''), such that the operation of a [[linear transformation|linear operator]] on a vector in the 2-plane always yields another vector in the same 2-plane. | |||
A particular case that has been studied is that in which the linear operator is an [[isometry]] ''M'' of the [[hypersphere]] (written ''S<sup>3</sup>'') represented within four-dimensional [[Euclidean space]]: | |||
:<math>M \; [ \mathbf{s} \; \mathbf{t} ] \; = \; [ \mathbf{s} \; \mathbf{t} ] \Lambda_\theta </math> | |||
where '''s''' and '''t''' are four-dimensional column vectors and Λ<sub>θ</sub> is a two-dimensional '''eigenrotation''' within the '''eigenplane'''. | |||
In the usual [[eigenvector]] problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom to multiply by an arbitrary non-zero [[rotation]]. | |||
This case is potentially physically interesting in the case that the [[shape of the universe]] is a [[multiply connected]] [[3-manifold]], since finding the [[angle]]s of the eigenrotations of a candidate isometry for [[shape of the universe|topological lensing]] is a way to [[scientific method|falsify]] such hypotheses. | |||
==See also== | |||
* [[Bivector]] | |||
* [[Plane of rotation]] | |||
==External links== | |||
*[http://arxiv.org/abs/astro-ph/0409694 possible relevance of eigenplanes] in [[physical cosmology|cosmology]] | |||
*[http://cosmo.torun.pl/GPLdownload/eigen/ GNU GPL software for calculating eigenplanes] | |||
[[Category:Linear algebra]] |
Revision as of 09:29, 11 April 2013
In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term eigenvector for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term eigenplane can be used to describe a two-dimensional plane (a 2-plane), such that the operation of a linear operator on a vector in the 2-plane always yields another vector in the same 2-plane.
A particular case that has been studied is that in which the linear operator is an isometry M of the hypersphere (written S3) represented within four-dimensional Euclidean space:
where s and t are four-dimensional column vectors and Λθ is a two-dimensional eigenrotation within the eigenplane.
In the usual eigenvector problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom to multiply by an arbitrary non-zero rotation.
This case is potentially physically interesting in the case that the shape of the universe is a multiply connected 3-manifold, since finding the angles of the eigenrotations of a candidate isometry for topological lensing is a way to falsify such hypotheses.